Topic 2.2 Representing Two Categorical Variables: construct and interpret two-way (contingency) tables and segmented or side-by-side bar graphs for two categorical variables.

What this topic is asking

The College Board (Topic 2.2) wants you to display two categorical variables together with a two-way (contingency) table and with segmented or side-by-side bar graphs, and to read the joint and marginal information these displays carry.

The two-way table

A complete table has every cell filled and every margin computed, with the row totals and column totals each summing to the grand total $n$. Filling in margins is not decoration: the marginal totals are needed to compute the proportions of the next topic, and a missing margin is a common reason a table-construction question loses marks.

Joint and marginal distributions

The table carries two kinds of information at once. The joint distribution is the set of cell counts (or cell proportions out of the grand total): it answers "how many are in both this row and this column?" The marginal distribution is given by the margins: a row total describes one variable on its own, ignoring the other. For example, the column totals tell you the overall split of the column variable across everyone, regardless of the row variable. Distinguishing joint from marginal is foundational, because conditional proportions (the next topic) build on both.

Segmented and side-by-side bar graphs

To display two categorical variables graphically, two bar-graph variants are standard. A segmented (stacked) bar graph draws one bar per category of the first variable, divided into segments whose heights show the proportions of the second variable within that category; comparing the segment patterns across bars reveals whether the variables are associated. A side-by-side (clustered) bar graph instead places the second variable's bars next to each other within each group, which can make exact comparisons easier when there are only a few categories. Both displays are usually drawn with relative frequencies (proportions) rather than raw counts when the groups have different sizes, so that the comparison is fair, exactly as in Unit 1. The choice between segmented and side-by-side is largely about readability: segmented bars emphasize parts of a whole, while side-by-side bars emphasize direct category-to-category comparison.

Reading association from the displays

The reason two-way tables and these bar graphs matter is that they reveal association between the two categorical variables. If the proportion preferring coffee is much higher among males than females, the segmented bars for the two sexes look different, and that visible difference is the association. If the variables were unrelated, the breakdown within each group would look the same. So when you read one of these displays, you are really asking "does the distribution of one variable change as I move across the categories of the other?" A strong exam answer names the variables, points to the differing proportions, and states the association in context, while remembering Topic 2.1's caution that an association in observational data is not proof of cause. The numerical version of this comparison, using conditional proportions, is exactly what Topic 2.3 formalises, so a clean two-way table here sets up the next topic directly.

Try this

Q1. In a two-way table, what do the column totals describe? [1 point]

• Cue. The marginal distribution of the column variable: that variable on its own, ignoring the row variable.

Q2. Why are relative frequencies preferred over counts in a segmented bar graph comparing two groups of different sizes? [2 points]

• Cue. Proportions put both groups on a per-total scale, so a difference in group size does not distort the visual comparison of the second variable's breakdown.